## Spot And Forward Yields

The spreadsheet entitled SPOTYA.XLS, found on the Online Learning Center (www.mhhe.com/bkm), can be used to estimate spot rates from coupon bonds and to calculate the forward rates for both single-year and multiyear bonds. The spreadsheet demonstrates a methodology to bootstrap spot rates from coupon bonds. The model sequentially solves for the spot rates that are associated with each of the periods. The methodology is similar to but slightly different from the regression methodology described in Section 15.6. Spot yields are derived from the yield curve of bonds that are selling at their par value, also referred to as the current coupon or "on-the-run" bond yield curve.

The spot rates for each maturity date are used to calculate the present value of each period's cash flow. The sum of these cash flows is the price of the bond. Given its price, the bond's yield to maturity can then be computed. If you were to err and use the yield to maturity of the on-the-run bond as the appropriate discount rate for each of the bond's coupon payments, you could find a significantly different price. That difference is calculated in the worksheet.

The spreadsheet uses the individual spot rates to estimate forward rates that we should observe under the Expectations Theory for the Term Structure of Interest Rates. Forward rate are estimated for one-year and multi-year bonds. The forward rates can be used to understand how market expectations would affect future rates.

 A B C D 1 Forward Rate Calculations 2 3 Spot Rate 1-yr for. 2-yr for. 4 Period Rate Rate 5 1 8.0000% 7.9792% 7.6770% 6 2 7.9896% 7.3756% 6.9188% 7 3 7.7846% 6.4639% 6.2612% 8 4 7.4529% 6.0588% 6.2864% 9 5 7.1726% 6.5145% 6.2611% 10 6 7.0626% 6.0084% 6.3863% 11 7 6.9114% 6.7656% 5.8388% 12 8 6.8932% 4.9201% 5.3305% 13 9 6.6721% 5.7425% 5.2994% 14 10 6.5788% 4.8582% 4.9254% 15 11 6.4212% 4.9926% 4.7620% 16 12 6.3014% 4.5320% 4.9682% 17 13 6.1642% 5.4062% 5.2220% 18 14 6.1099% 5.0381% 4.9448% 19 15 6.0381% 4.8516% 4.7551% 20 16 5.9636% 4.6586% 4.5590% 21 17 5.8864% 4.4594% 4.9627% 22 18 5.8066% 5.4684% 5.4353% 23 19 5.7887% 5.4022% 24 20 5.7694%

CHAPTER 15 The Term Structure of Interest Rates 471

separate payments from many bonds into portfolios with common maturity dates. By determining the price of each of these "zeros" we can calculate the yield to that maturity date for a single-payment security and thereby construct the pure yield curve.

As a simple example of this technique, suppose that we observe an 8% coupon bond making semiannual payments with one year until maturity, selling at \$986.10, and a 10% coupon bond, also with a year until maturity, selling at \$1,004.78. To infer the short rates for the next two six-month periods, we first attempt to find the present value of each coupon payment taken individually, that is, treated as a mini-zero-coupon bond. Call d1 the present value of \$1 to be received in half a year and d2 the present value of a dollar to be received in one year. (The d stands for discounted values; therefore, d1 = 1/(1 + r1), where r1 is the short rate for the first six-month period.) Then our two bonds must satisfy the simultaneous equations

In each equation the bond's price is set equal to the discounted value of all of its remaining cash flows. Solving these equations we find that d1 = .95694 and d2 = .91137. Thus if r1 is the short rate for the first six-month period, then d1 = 1/(1 + r1) = .95694, so that r1 = .045, and d2 = 1/[(1 + r1)(1 + f2)] = 1/[(1.045)(1 + f2)] = .91137, so that f2 = .05. Thus the two short rates are shown to be 4.5% for the first half-year period and 5% for the second.

CONCEPT CHECK ^ QUESTION 8

A T-bill with six-month maturity and \$10,000 face value sells for \$9,700. A one-year maturity T-bond paying semiannual coupons of \$40 sells for \$1,000. Find the current six-month short rate and the forward rate for the following six-month period.

When we analyze many bonds, such an inference procedure is more difficult, in part because of the greater number of bonds and time periods, but also because not all bonds give rise to identical estimates for the discounted value of a future \$1 payment. In other words, there seem to be apparent error terms in the pricing relationship.3 Nevertheless, treating these errors as random aberrations, we can use a statistical approach to infer the pattern of forward rates embedded in the yield curve.

To see how the statistical procedure would operate, suppose that we observe many coupon bonds, indexed by i, selling at prices Pi. The coupon and/or principal payment (the cash flow) of bond i at time t is denoted CFit, and the present value of a \$1 payment at time t, which is the implied price of a zero-coupon bond that we are trying to determine, is denoted dt. Then for each bond we may write the following:

Pi

= d1CF11 -

1- d2CFi2 -

- d3CF13 + . .

. . + ei

P2

= diCF2i -

d2CF22

d3CF23 - .

. . + e2

P3

= d1CF31 -

d2CF32

h d3CF33 + .

Each line of equation system 15.7 equates the price of the bond to the sum of its cash flows, discounted according to time until payment. The last term in each equation, ei, represents

3 We will consider later some of the reasons for the appearance of these error terms.

PART IV Fixed-Income Securities the error term that accounts for the deviations of a bond's price from the prediction of the equation.

Students of statistics will recognize that equation 15.7 is a simple system of equations that can be estimated by regression analysis. The dependent variables are the bond prices, the independent variables are the cash flows, and the coefficients dt are to be estimated from the observed data.4 The estimates of dt are our inferences of the present value of \$1 to be paid at time t. The pattern of dt for various times to payment is called the discount function, because it gives the discounted value of \$1 as a function of time until payment. From the discount function, which is equivalent to a list of zero-coupon bond prices for various maturity dates, we can calculate the yields on pure zero-coupon bonds. We would use Treasury securities in this procedure to avoid complications arising from default risk.

Before leaving the issue of the measurement of the yield curve, it is worth pausing briefly to discuss the error terms. Why is it that all bond prices do not conform exactly to a common discount function that sets price equal to present value? Two of the reasons relate to factors not accounted for in the regression analysis of equation 15.7: taxes and options associated with the bond.

Taxes affect bond prices because investors care about their after-tax return on investment. Therefore, the coupon payments should be treated as net of taxes. Similarly, if a bond is not selling at par value, the IRS may impute a "built-in" interest payment by amortizing the difference between the price and the par value of the bond. These considerations are difficult to capture in a mathematical formulation because different individuals are in different tax brackets, meaning that the net-of-tax cash flows from a given bond depend on the identity of the owner. Moreover, the specification of equation 15.7 implicitly assumes that the bond is held until maturity: It discounts all the bond's coupon and principal payments. This, of course, ignores the investor's option to sell the bond before maturity and so to realize a different stream of income from that described by equation 15.7. Moreover, it ignores the investor's ability to engage in tax-timing options. For example, an investor whose tax bracket is expected to change over time may benefit by realizing capital gains during the period when the tax rate is the lowest.

Another feature affecting bond pricing is the call provision. First, if the bond is callable, how do we know whether to include in equation 15.7 coupon payments in years following the first call date? Similarly, the date of the principal repayment becomes uncertain. More important, one must realize that the issuer of the callable bond will exercise the option to call only when it is profitable to do so. Conversely, the call provision is a transfer of value away from the bondholder who has "sold" the option to call to the bond issuer. The call feature therefore will affect the bond's price and introduce further error terms in the simple specification of equation 15.7.

Finally, we must recognize that the yield curve is based on price quotes that often are somewhat inaccurate. Price quotes used in the financial press may be stale (i.e., out of date), even if only by a few hours. Moreover, they may not represent prices at which dealers actually are willing to trade.

### SUMMARY

1. The term structure of interest rates refers to the interest rates for various terms to maturity embodied in the prices of default-free zero-coupon bonds.

4 In practice, variations of regression analysis called "splining techniques" are usually used to estimate the coefficients. This method was first suggested by McCulloch in the following two articles: J. Huston McCulloch, "Measuring the Term Structure of Interest Rates," Journal of Business 44 (January 1971); and "The Tax Adjusted Yield Curve," Journal of Finance 30 (June 1975).

### CHAPTER 15 The Term Structure of Interest Rates

2. In a world of certainty all investments must provide equal total returns for any investment period. Short-term holding-period returns on all bonds would be equal in a risk-free economy, and all equal to the rate available on short-term bonds. Similarly, total returns from rolling over short-term bonds over longer periods would equal the total return available from long-maturity bonds.

3. The forward rate of interest is the break-even future interest rate that would equate the total return from a rollover strategy to that of a longer-term zero-coupon bond. It is defined by the equation

where n is a given number of periods from today. This equation can be used to show that yields to maturity and forward rates are related by the equation

(1 + y„)" = (1 + r1)(1 + f2)(1 + f3) . . . (1 + fn)

4. A common version of the expectations hypothesis holds that forward interest rates are unbiased estimates of expected future interest rates. However, there are good reasons to believe that forward rates differ from expected short rates because of a risk premium known as a liquidity premium. A liquidity premium can cause the yield curve to slope upward even if no increase in short rates is anticipated.

5. The existence of liquidity premiums makes it extremely difficult to infer expected future interest rates from the yield curve. Such an inference would be made easier if we could assume the liquidity premium remained reasonably stable over time. However, both empirical and theoretical insights cast doubt on the constancy of that premium.

6. A pure yield curve could be plotted easily from a complete set of zero-coupon bonds. In practice, however, most bonds carry coupons, payable at different future times, so that yield-curve estimates are often inferred from prices of coupon bonds. Measurement of the term structure is complicated by tax issues such as tax timing options and the different tax brackets of different investors.

### KEY TERMS

term structure of interest rates short interest rate yield curve spot rate forward interest rate liquidity premium expectations hypothesis liquidity preference theory market segmentation theory preferred habitat theory term premiums

This site is a good source for current rates, current and past yield curves, and discussions of how the shape of the yield curve can affect economic performance. It also has a summary of current economic factors influencing rates.

http://www.smartmoney.com/bonds/

The sites listed below contain price and yield curve information as well as the ability to chart Treasury security prices over time.

http://www.bondresources.com/

http://www.bloomberg.com/markets

http://www.investinginbonds.com

WEBSITES

Bodie-Kane-Marcus: Investments, Fifth Edition

IV. Fixed-Income Securities

15. The Term Structure of Interest Rates

 WEBSITES Historical information on interest rates and other economic factors are available in the Federal Reserve Economic Data Base (FRED) at the address shown below. Data in FRED can be downloaded in spreadsheet format. http://www.stls.frb.ors/

PROBLEMS

1. Briefly explain why bonds of different maturities have different yields in terms of the (1) expectations, (2) liquidity, and (3) segmentation hypotheses. Briefly describe the implications of each of the three hypotheses when the yield curve is (1) upward sloping and (2) downward sloping.

2. Which one of the following statements about the term structure of interest rates is true?

a. The expectations hypothesis indicates a flat yield curve if anticipated future short-term rates exceed current short-term rates.

b. The expectations hypothesis contends that the long-term rate is equal to the anticipated short-term rate.

c. The liquidity premium theory indicates that, all else being equal, longer maturities will have lower yields.

d. The market segmentation theory contends that borrowers and lenders prefer particular segments of the yield curve.

3. The differences between short and forward rates are most closely associated with which one of the following explanations of the term structure of interest rates?

a. Expectations hypothesis.

c. Preferred habitat hypothesis.

d. Segmented market theory.

4. Under the expectations hypothesis, if the yield curve is upward sloping, the market must expect an increase in short-term interest rates. True/false/uncertain? Why?

5. Under the liquidity preference theory, if inflation is expected to be falling over the next few years, long-term interest rates will be higher than short-term rates. True/false/ uncertain? Why?

6. The following is a list of prices for zero-coupon bonds of various maturities. Calculate the yields to maturity of each bond and the implied sequence of forward rates.

 Maturity (Years) Price of Bond (\$) 1 943.40 2 898.47 3 847.62 4 792.16

7. Assuming that the expectations hypothesis is valid, compute the expected price path of the four-year bond in problem 6 as time passes. What is the rate of return of the bond in each year? Show that the expected return equals the forward rate for each year.

8. Suppose the following table shows yields to maturity of U.S. Treasury securities as of January 1, 1996:

Bodie-Kane-Marcus: I IV. Fixed-Income I 15. The Term Structure of I I © The McGraw-Hill

Investments, Fifth Edition Securities Interest Rates Companies, 2001

CHAPTER 15 The Term Structure of Interest Rates

 Term to Maturity (Years) Yield to Maturity 1 3.50% 2 4.50 3 5.00 4 5.50 5 6.00 10 6.60

a. Based on the data in the table, calculate the implied forward one-year rate of interest at January 1, 1999.

b. Describe the conditions under which the calculated forward rate would be an unbiased estimate of the one-year spot rate of interest at January 1, 1999.

c. Assume that one year earlier, at January 1, 1995, the prevailing term structure for U.S. Treasury securities was such that the implied forward one-year rate of interest at January 1, 1999, was significantly higher than the corresponding rate implied by the term structure at January 1, 1996. On the basis of the pure expectations theory of the term structure, briefly discuss two factors that could account for such a decline in the implied forward rate.

9. Would you expect the yield on a callable bond to lie above or below a yield curve fitted from noncallable bonds?

10. The six-month Treasury bill spot rate is 4%, and the one-year Treasury bill spot rate is 5%. The implied six-month forward rate for six months from now is:

11. The tables below show, respectively, the characteristics of two annual-pay bonds from the same issuer with the same priority in the event of default, and spot interest rates. Neither bond's price is consistent with the spot rates. Using the information in these tables, recommend either bond A or bond B for purchase. Justify your choice.

 Bond A Bond B Coupons Annual Annual Maturity 3 years 3 years Coupon rate 10% 6% Yield to maturity 10.65% 10.75% Price 98.40 88.34
 Term (Years) Spot Rates (Zero Coupon) 1 5% 2 8 3 11

PART IV Fixed-Income Securities

12. The current yield curve for default-free zero-coupon bonds is as follows:

 Maturity (Years) YTM 1 10% 2 11 3 12

13, a. What are the implied one-year forward rates?

b. Assume that the pure expectations hypothesis of the term structure is correct. If market expectations are accurate, what will the pure yield curve (that is, the yields to maturity on one- and two-year zero coupon bonds) be next year?

c. If you purchase a two-year zero-coupon bond now, what is the expected total rate of return over the next year? What if you purchase a three-year zero-coupon bond? (Hint: Compute the current and expected future prices.) Ignore taxes.

d. What should be the current price of a three-year maturity bond with a 12% coupon rate paid annually? If you purchased it at that price, what would your total expected rate of return be over the next year (coupon plus price change)? Ignore taxes.

The term structure for zero-coupon bonds is currently:

 Maturity (Years) YTM 1 4% 2 5 3 6

Next year at this time, you expect it to be:

 Maturity (Years) YTM 1 5% 2 6 3 7

a. What do vou expect the rate of return to be over the coming year on a three-year zero-coupon bond?

b. Under the expectations theory, what yields to maturity does the market expect to observe on one- and two-year zeros next year? Is the market's expectation of the return on the three-year bond greater or less than yours?

14. The yield to maturity on one-year zero-coupon bonds is currently 7%; the YTM on two-year zeros is 8%. The Treasury plans to issue a two-year maturity coupon bond, paying coupons once per year with a coupon rate of 9%. The face value of the bond is \$100.

a. At what price will the bond sell?

b. What will the yield to maturity on the bond be?

c. If the expectations theory of the yield curve is correct, what is the market expectation of the price that the bond will sell for next year?

d. Recalculate your answer to (c) if you believe in the liquidity preference theory and you believe that the liquidity premium is 1%.

### CHAPTER 15 The Term Structure of Interest Rates

15. A portfolio manager at Superior Trust Company is structuring a fixed-income portfolio to meet the objectives of a client. This client plans on retiring in 15 years and wants a substantial lump sum at that time. The client has specified the use of AAA-rated securities.

The portfolio manager compares coupon U.S. Treasuries with zero-coupon stripped U.S. Treasuries and observes a significant yield advantage for the stripped bonds:

 Term (Years) Coupon U.S. Treasuries Zero-Coupon Stripped U.S. Treasuries 3 5.50% 5.80% 5 6.00 6.60 7 6.75 7.25 10 7.25 7.60 15 7.40 7.80 30 7.75 8.20

Briefly discuss why zero-coupon stripped U.S. Treasuries could yield more than coupon U.S. Treasuries with the same final maturity. 16. Below is a list of prices for zero-coupon bonds of various maturities.

 Maturity (Years) Price of \$1,000 Par Bond (Zero Coupon) 1 943.40 2 873.52 3 816.37

a. An 8.5% coupon \$1,000 par bond pays an annual coupon and will mature in three years. What should the yield to maturity on the bond be?

b. If at the end of the first year the yield curve flattens out at 8%, what will be the one-year holding-period return on the coupon bond?

17. Prices of zero-coupon bonds reveal the following pattern of forward rates:

 Year Forward Rate 1 5% 2 7 3 8

In addition to the zero-coupon bond, investors also may purchase a three-year bond making annual payments of \$60 with par value \$1,000.

a. What is the price of the coupon bond?

b. What is the yield to maturity of the coupon bond?

c. Under the expectations hypothesis, what is the expected realized compound yield of the coupon bond?

d. If you forecast that the yield curve in one year will be flat at 7%, what is your forecast for the expected rate of return on the coupon bond for the one-year holding period?

18. The shape of the U.S. Treasury yield curve appears to reflect two expected Federal Reserve reductions in the Federal Funds rate. The current short-term interest rate is 5%.

Bodie-Kane-Marcus: Investments, Fifth Edition

IV. Fixed-Income Securities

15. The Term Structure of Interest Rates

### PART IV Fixed-Income Securities

The first reduction of approximately 50 basis points (bp) is expected six months from now and the second reduction of approximately 50 bp is expected one year from now. The current U.S. Treasury term premiums are 10 bp per year for each of the next 3 years (out through the 3-year benchmark).

However, the market also believes that the Federal Reserve reductions will be reversed in a single 100 bp increase in the Federal Funds rate 21/2 years from now. You expect term premiums to remain 10 bp per year for each of the next 3 years (out through the 3-year benchmark).

Describe or draw the shape of the Treasury yield curve out through the 3-year benchmark. State which term structure theory supports the shape of the U.S. Treasury yield curve you've described.

19. You observe the following term structure:

 Effective Annual YTM 1-year zero-coupon bond 6.1% 2-year zero-coupon bond 6.2 3-year zero-coupon bond 6.3 4-year zero-coupon bond 6.4

a. If you believe that the term structure next year will be the same as today's, will the one-year or the four-year zeros provide a greater expected one-year return?

b. What if you believe in the expectations hypothesis?

20. U.S. Treasuries represent a significant holding in many pension portfolios. You decide to analyze the yield curve for U.S. Treasury notes.

a. Using the data in the table below, calculate the five-year spot and forward rates assuming annual compounding. Show your calculations.

 Par Coupon Calculated Calculated Years to Maturity Yield to Maturity Spot Rates Forward Rates 1 5.00 5.00 5.00 2 5.20 5.21 5.42 3 6.00 6.05 7.75 4 7.00 7.16 10.56 5 7.00 ? ?

b. Define and describe each of the following three concepts:

i. Yield to maturity.

ii. Spot rate.

iii. Forward rate.

### Explain how these concepts are related.

c. You are considering the purchase of a zero-coupon U.S. Treasury note with four years to maturity. Based on the above yield-curve analysis, calculate both the expected yield to maturity and the price for the security. Show your calculations.

21. The yield to maturity (YTM) on one-year zero-coupon bonds is 5% and the YTM on two-year zeros is 6%. The yield to maturity on two-year-maturity coupon bonds with

CHAPTER 15 The Term Structure of Interest Rates coupon rates of 12% (paid annually) is 5.8%. What arbitrage opportunity is available for an investment banking firm? What is the profit on the activity?

22. Suppose that a one-year zero-coupon bond with face value \$100 currently sells at \$94.34, while a two-year zero sells at \$84.99. You are considering the purchase of a two-year-maturity bond making annual coupon payments. The face value of the bond is \$100, and the coupon rate is 12% per year.

a. What is the yield to maturity of the two-year zero? The two-year coupon bond?

b. What is the forward rate for the second year?

c. If the expectations hypothesis is accepted, what are (1) the expected price of the coupon bond at the end of the first year and (2) the expected holding-period return on the coupon bond over the first year?

d. Will the expected rate of return be higher or lower if you accept the liquidity preference hypothesis?

23. Suppose that the prices of zero-coupon bonds with various maturities are given in the following table. The face value of each bond is \$1,000.

 Maturity (Years) Price 1 925.93 2 853.39 3 782.92 4 715.00 5 650.00

a. Calculate the forward rate of interest for each year.

b. How could you construct a one-year forward loan beginning in year 3? Confirm that the rate on that loan equals the forward rate.

c. Repeat (b) for a one-year forward loan beginning in year 4.

24. Continue to use the data in the preceding problem. Suppose that you want to construct a two-year maturity forward loan commencing in three years.

a. Suppose that you buy today one three-year maturity zero-coupon bond. How many five-year maturity zeros would you have to sell to make your initial cash flow equal to zero?

b. What are the cash flows on this strategy in each year?

c. What is the effective two-year interest rate on the effective three-year-ahead forward loan?

d. Confirm that the effective two-year interest rate equals (1 + f4) X (1 + f5) - 1. You therefore can interpret the two-year loan rate as a two-year forward rate for the last two years. Alternatively, show that the effective two-year forward rate equals

25. The following are the current coupon yields to maturity and spot rates of interest for six U.S. Treasury securities. Assume all securities pay interest annually.

PART IV Fixed-Income Securities

 Term to Maturity Current Coupon Yield to Maturity Spot Rate of Interest 1-Year Treasury 5.25% 5.25% 2-Year Treasury 5.75 5.79 3-Year Treasury 6.15 6.19 5-Year Treasury 6.45 6.51 10-Year Treasury 6.95 7.10 30-Year Treasury 7.25 7.67

Compute, under the expectations theory, the two-year implied forward rate three years from now, given the information provided in the preceding table. State the assumption underlying the calculation of the implied forward rate.

### SOLUTIONS TO CONCEPT CHECKS

1. The bond sells today for \$683.18 (from Table 15.2). Next year, it will sell for \$1,000/[(1.10)(1.11)(1.11)] = \$737.84, for a return 1 + r = 737.84/683.18 = 1.08, or r = 8%.

2. The n-period spot rate is the yield to maturity on a zero-coupon bond with a maturity of n periods. The short rate for period n is the one-period interest rate that will prevail in period n. Finally, the forward rate for period n is the short rate that would satisfy a "break-even condition" equating the total returns on two n-period investment strategies. The first strategy is an investment in an n-period zero-coupon bond; the second is an investment in an n - 1 period zero-coupon bond "rolled over" into an investment in a one-period zero. Spot rates and forward rates are observable today, but because interest rates evolve with uncertainty, future short rates are not. In the special case in which there is no uncertainty in future interest rates, the forward rate calculated from the yield curve would equal the short rate that will prevail in that period.

4. The risk premium will be zero.

5. If issuers prefer to issue long-term bonds, they will be willing to accept higher expected interest costs on long bonds over short bonds. This willingness combines with investors' demands for higher rates on long-term bonds to reinforce the tendency toward a positive liquidity premium.

6. If r4 equaled 9.66%, then the four-year bond would sell for \$1,000/ [(1.08)(1.10)(1.11)(1.0966)] = \$691.53. The yield to maturity would satisfy the equation 691.53(1 + y4)4 = 1,000, or y4 = 9.66%. At a lower value of r4, the bond would sell for a higher price and offer a lower yield. At a higher value of r4, the yield would be greater.

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### Responses

• MARI
What do you expect the rate of return to be over the coming year on a 3year zerocoupon bond?
7 years ago
• franziska
What is the face value of the SF bond if the investments are equivalent at the spot rate?
6 years ago